Simultaneous Determination of Thermal and Mutual Diffusivity ofBinary Mixtures of n‑Octacosane with Carbon Monoxide, Hydrogen,and Water by Dynamic Light ScatteringAndreas Heller,† Thomas M. Koller,† Michael H. Rausch,†,‡ Matthieu S. H. Fleys,§ A. N. Rene Bos,§Gerard P. van der Laan,§ Zoi A. Makrodimitri,∥ Ioannis G. Economou,∥,⊥ and Andreas P. Froba*,†,‡

†Erlangen Graduate School in Advanced Optical Technologies (SAOT), University of Erlangen-Nuremberg, Paul-Gordan-Straße 6,D-91052 Erlangen, Bavaria, Germany‡Department of Chemical and Biological Engineering, Institute of Engineering Thermodynamics, University of Erlangen-Nuremberg,Am Weichselgarten 8, D-91058 Erlangen, Bavaria, Germany§Shell Global Solutions International B.V., Grasweg 31, 1031 HW Amsterdam, The Netherlands∥National Centre for Scientific Research “Demokritos”, Institute of Advanced Materials, Physicochemical Processes, Nanotechnologyand Microsystems, Molecular Thermodynamics and Modelling of Materials Laboratory, GR-15310 Aghia Paraskevi Attikis, Greece⊥Chemical Engineering Program, Texas A&M University at Qatar, Education City, PO Box 23874, Doha, Qatar

ABSTRACT: It is demonstrated that thermal and mutual diffusivities of binary mixtures ofn-octacosane (n-C28H58) with carbon monoxide (CO), hydrogen (H2), and water (H2O)are simultaneously accessible by dynamic light scattering (DLS). As the light-scatteringsignals originating from thermal and concentration fluctuations appear in similar time scales,different data evaluation strategies were tested to achieve minimum uncertainties in theresulting transport properties. To test the agreement of the respective theoretical modelwith the DLS signals in the regression, an improved multifit procedure is introduced. Withthe selected data evaluation strategy, uncertainties of 4 to 15% and 4 to 30% in the thermaland mutual diffusivities, respectively, could be obtained for the binary mixtures. The mutualdiffusivities for the mixtures measured at temperatures ranging from 398 to 523 K andpressures of 5 to 30 bar at saturation conditions are in good agreement with moleculardynamics simulations and data from the literature.

■ INTRODUCTIONThe mutual diffusivity of liquids containing dissolved gases isof technological interest in different industrial fields ofchemical and energy engineering.1−3 The characterization ofsuch systems over a wide region of thermodynamic statesrepresents a metrological challenge. In the literature, manyconventional measurement techniques capable of determiningmutual diffusivities of liquids containing dissolved gases canbe found, all of them having their own restrictions. Forexample, the Taylor dispersion (TD) can be used todetermine the mutual diffusivity by studying the concentrationdistribution of a solute dispersed in a flowing solvent stream.To the best of our knowledge, up to now the TD has beenapplied only to systems at infinite dilution.4−6 Furthermore,the transient gravimetric7,8 and open tube methods9−11 can berepresentatively mentioned. They make use of macroscopicgradients in concentration which must be large enough to giverise to a measurable effect but small enough to cause onlyvery little perturbation in the system under investigation.Dynamic light scattering (DLS), which is used in the presentstudy, is restricted to the hydrodynamic regime where thelinearized Navier−Stokes equations are valid. For thedetermination of mutual diffusivities in binary mixtures, thesolute concentration has to be large enough to obtain

detectable DLS signals that can be attributed to the moleculardiffusion process. On the other hand, DLS allows thedetermination of transport and other thermophysical proper-ties in macroscopic thermodynamic equilibrium in an absoluteway over a wide range of thermodynamic states.12,13

DLS was recently applied for the determination of mutualdiffusivities of mixtures of ionic liquids with molecularsolvents.14,15 In the present study, the DLS technique wasutilized for the first time to investigate mutual and thermaldiffusivities of liquids containing dissolved gases. Bothproperties are related to the line width of the Rayleighcomponent of the spectrum of scattered light which arisesfrom temperature and concentration fluctuations. From thecalculated time-dependent correlation function (CF) of theintensity of scattered light, the mutual and thermal diffusivitycan be evaluated.16,17

For mixtures exhibiting mutual and thermal diffusivities onthe same order of magnitude, the corresponding hydro-dynamic modes appear on the same time scale in the CF.Investigations on a methane−ethane mixture in the near-

Received: January 10, 2014Revised: February 14, 2014Published: March 20, 2014

Article

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© 2014 American Chemical Society 3981 dx.doi.org/10.1021/jp500300y | J. Phys. Chem. B 2014, 118, 3981−3990

critical region showed that a simultaneous determination ofmutual and thermal diffusivity is possible in the case whenboth transport properties expose similar values.18 In thepresent study, it is demonstrated that the mutual and thermaldiffusivity for liquids containing dissolved gases are alsosimultaneously accessible by DLS. As a model system, n-octacosane (n-C28H58) containing carbon monoxide (CO),hydrogen (H2), or water (H2O) was chosen. Information onthe mutual diffusivity of such systems is required for afundamental understanding of the Fischer−Tropsch syn-thesis.1 Earlier studies4,19−21 have shown that the mutualdiffusivity of n-C28H58 mixtures with dissolved CO, H2, orH2O is on the same order of magnitude as the thermaldiffusivity of n-C28H58, which should allow for a simultaneousdetermination of both transport properties by DLS. In thiscase, the key to precisely accessing mutual and thermaldiffusivity from the recorded CFs containing superimposedsignals is the applied data evaluation strategy. A new dataevaluation procedure for the chosen systems probed atprocess-relevant temperatures ranging from 398 to 523 Kand pressures from 5 to 30 bar as well as the correspondingdata are presented and discussed.

■ METHODThe principles of DLS for the determination of transport andother thermophysical properties are described elsewhere indetail.12,16,17 Here, only the essential features relevant for thepresent study are presented.The decay of the two hydrodynamic modes present in

binary fluid mixtures follows the same laws that are valid formacroscopic systems. Thus, the decay of temperature andconcentration fluctuations is governed by the thermaldiffusivity and the mutual diffusivity, respectively. Informationabout these equilibration processes can be derived by thetemporal analysis of the scattered light intensity by DLS. Themean decay times of the two hydrodynamic modes arecalculated from the time-dependent CF of the intensity ofscattered light. Taking into account that in a realisticexperimental situation reference light, e.g., stray light fromthe windows of the sample cell, is superimposed coherentlyon the scattered light from the sample, the intensitycorrelation function G(2)(τ) takes the form

Here, ILO is the intensity of the reference light or the so-called local oscillator and It and Ic denote the scattering lightintensities caused by temperature and concentration fluctua-tions, respectively. In contrast to the homodyne term and thecross term, which are due to the scattered light from thesample alone, the heterodyne term is the result of theinterference of the scattered and the reference light. This termis characterized by decay times of fluctuations in temperatureand concentration, τC,t and τC,c. The decay time τC,t , which isequivalent to the mean lifetime of the temperaturefluctuations observed, is related to the thermal diffusivity a by

τ=a

q1

C,t2

(2)

Equivalently, from the characteristic decay time τC,c , whichis equivalent to the mean lifetime of the concentration

fluctuations observed, the mutual diffusivity D12 can bedetermined by

τ=D

q1

12C,c

2(3)

In eqs 2 and 3, q is the modulus of the scattering vector

πλ

= | − | ≅ Θ = Θq k k kn

2 sin( /2)4

sin( /2)i s i s0

s(4)

which is defined by the difference of the wave vectors ofincident and scattered light, ki and ks, respectively. Assumingelastic scattering (ki ≅ ks), the modulus of the scatteringvector is given in terms of the fluid refractive index n, thelaser wavelength in vacuo λ0, and the scattering angle Θs. Forsmall scattering angles, an approximation for the modulus ofthe scattering vector can be deduced, q ≅ (2π/λ0) sinΘi ,where Θi denotes the incident angle. In this case, noinformation on the refractive index of the fluid is needed.Because of the large number of parameters in the

correlation function, eq 1, it is very difficult to obtain thedecay times τC,t and τC,c. The problem is simplified ifheterodyne conditions can be arranged, i.e., It ≪ ILO and Ic ≪ILO. The normalized correlation function is reduced to a sumof two exponentials reflecting the mean lifetime of temper-ature and concentration fluctuations according to

τ τ τ τ τ= + − + −g b b b( ) exp( / ) exp( / )(2)0 t C,t c C,c (5)

The experimental constants b0 , bt , and bc include both thecorresponding terms from eq 1 and effects caused by theimperfect signal collection due to incoherent background andthe finite detector area.If thermal and mutual diffusivity of an investigated system

are on the same order of magnitude, the two correspondinghydrodynamic modes present in binary fluid mixtures areobservable on the same time scale. Whether it is possible toresolve both signals of the correlation function given by eq 5is restricted mainly by the ratio of the scattering intensities It/Ic , which depends on the relative difference of the refractiveindices of the two components and their concentration. Ingeneral, the simultaneous determination of both signals inbinary mixtures is possible if the refractive indices of the twocomponents are comparable. In the case of a large differencebetween the refractive indices, the scattered light intensityarising from concentration fluctuations dominates thecorrelation function. If the refractive indices of the purecomponents of the mixtures match, no information on theconcentration fluctuations can be found in the scattered light.

■ EXPERIMENTAL SECTIONA scheme of the optical setup used in this study is illustratedin Figure 1. A frequency-doubled Nd:YVO4 laser sourceoperated at 532 nm and 500 mW output power is focusedinto the sample cell (SC) by mirrors (M) and a lens (L) witha focal length of f = 2 m. The intensity of the main beamindicated by the solid line can be adjusted by a combinationof a half-wave plate (λ/2 retardation) and a polarization beamsplitter (PBS). With the setup, scattering angles ranging from−9° to +9° can be realized, where the approximation for qcan be applied. The incident angle Θi is adjusted by themirror M6 and measured with a rotational table using theautocollimation technique with an uncertainty of ±0.01°.

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Two stops (P1, P2) with a distance of about 1 m areinstalled behind the sample cell defining the detectiondirection. To achieve heterodyne conditions, a local oscillatorindicated by the dashed line is superimposed with thescattered light. Its intensity is adjusted by a gray filter (GF).The resulting signal is detected by two photomultiplier tubes(PMTs). During the experiment, the pseudocross correlationfunction is calculated by a single-tau correlator featuring 255equally spaced channels and a multitau correlator simulta-neously.The sample cell with a total inner volume of 40 mL is

made of hydrogen-resistant steel and discloses four opticalaccesses, which are sealed by an O-ring system. Firstinvestigations showed that the sample was contaminatedwith particles stemming from the corrosion of the cellmaterial. To render the surface of the sample cell inert, it wasgalvanically gold plated. However, because of strong thermaland mechanical stress induced by the measurement andcleaning procedure, the gold layer became unstable after sometime. A more chemically stable nickel coating countered theeffects of mechanical stress. At elevated temperatures,however, the nickel reacted with CO to form carbonylcomplexes. Taking these aspects into account, the measure-ments for pure n-C28H58 and its mixtures with CO werecarried out with the uncoated and gold-coated sample cell.Investigations for the mixtures with H2 and H2O were carriedout with the nickel-coated sample cell.A maximum sample temperature of 573 K can be achieved

by resistance heating with a temperature stability of betterthan 3 mK. Two calibrated Pt 100 Ω resistance probes areused to measure the cell temperature with an absoluteuncertainty (k = 2) of less than 10 mK. The temperaturecontrol loop is realized with a temperature probe placed inthe wall of the cell close to the resistance heating. To measurethe temperature of the fluid, the second probe is placed insidethe cell material, located close to the fluid. The sample cellcan be closed by two bellow-type valves. It is connected to amanifold allowing evacuation, flushing, and dosing of thesample gases (Figure 2). An additional resistance heatingsystem is installed at the tubing directly connected to thesample cell to eliminate convection. The correspondingtemperature control loop is probed by a further calibratedtemperature sensor with an absolute uncertainty of less than10 mK. Here, temperature stability better than 0.1 K wasachieved.

The manifold including the sample cell is installed within afume hood. CO or H2 is loaded into the gas buffer after thesystem was flushed with nitrogen (N2) and evacuated with anoil-sealed vacuum pump (0.5 mbar) several times. Themanifold and the measurement cell were designed to handlepressures up to 80 bar. The pressure in the sample cell isrecorded by a pressure transducer with an uncertainty of lessthan 0.05%.The investigated n-C28H58 was purchased from Alfa Aesar

GmbH & Co. KG with a purity of 99% by mass. At ambientconditions, the sample is solid. The liquefied sample showsparticle-like impurities presumably resulting from themanufacturing process. To obtain particle-free samples asnecessary for DLS experiments, the n-C28H58 was filtered witha syringe filter with 200 nm pore size at about 353 K. CO andH2 were provided by Linde AG with purities of 99.997 vol %and 99.9999 vol %. About 30 mL of liquid filtered alkanewere filled into the sample cell at a temperature of 353 K. Atthis temperature, the filled sample cell is connected to themanifold. Because of the low vapor pressure of n-C28H58 at353 K, the remaining air in the sample cell can be removed byan oil-sealed vacuum pump (0.5 mbar) before it is filled withCO or H2. After equilibration of the two-phase system for atleast 24 h, first measurements were carried out. After thattime, no significant changes in the DLS signals could beobserved, indicating a steady state of the investigated system.For the preparation of the mixtures with H2O, 2−5 mL of

distilled water were added to the sample cell containing liquidn-C28H58. After the cell was assembled and connected to themanifold, the remaining air in the cell was removed by an oil-sealed vacuum pump. The investigations for the mixtures withCO and H2 were carried out in the liquid phase in thepresence of a gas phase above. For the mixture with H2O, asecond liquid phase mainly consisting of H2O could beobserved directly after filling at 353 K. This water phase couldnot be observed after heating the sample cell to temperatureshigher than 448 K. Nevertheless, the system can be specifiedregarding its composition, assuming the recorded pressurecorresponds to the saturated pressure of water. To specify theconcentration of the dissolved solute in the n-alkane, thecomposition of each mixture was estimated from solubilitydata,22−24 the recorded temperature, and the measuredsaturation pressure. Solubility data for mixtures with COand H2 are given from 348.2 to 423.2 K and had to beextrapolated to temperatures up to 523 K. This extrapolation

Figure 1. Optical setup. Figure 2. Manifold for dosing CO or H2 into the sample cell.

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was performed by describing the solubility data with a surfacepolynomial with pressure and temperature dependence toestimate the composition at our experimental conditions. Theerror in each calculated mole fraction is estimated to besmaller than 0.02. The composition of the mixture with H2Owas calculated based on pseudo-Henry constants reported byBreman et al.24 in the temperature range from 428.8 to 519.0K. According to the authors, the pseudo-Henry constantdeviates by less than 10% from the real Henry constant; thisresults in an estimated error in the calculated mole fraction ofless than 0.02. The calculated compositions for the CO andH2 mixtures based on solubility data reported by Srivatsan etal.22 and Park et al.23 were compared with composition dataderived from pseudo-Henry constants reported by Breman etal.24 Between both estimates of the mixture composition, amaximum deviation of 0.04 in the mole fraction could beobserved over the whole temperature range.During the measurements, the bellow-type valves which

connect the manifold with the measurement cell are closed toensure steady-state conditions. Unpreventable diffusion of gasthrough the soft sealings of the optical accesses causedongoing pressure loss. For the mixture with H2 at atemperature of 473.04 K and an absolute pressure of about32 bar, a maximum pressure drop of 1 bar h−1 was observed.According to solubility data, however, such a pressure lossresults in a change of the H2 mole fraction of less than 0.006per hour. At lower temperatures, this pressure drop reduces toless than 0.2 bar h−1. The observed pressure drop for themixtures with CO or H2O was less than 0.2 bar h−1 at allinvestigated temperatures. This described pressure loss causesconvective effects in the sample which can be one reason fordisturbances observable in the DLS signals. However, thesedisturbing signals are present in a different time range thanthe signals related to molecular or thermal diffusion processes.For each defined temperature, six measurements at differentscattering angles ranging from −6° to +6° were performed.For each single measurement lasting from 10 to 120 min,depending on the light scattering intensities, the pressure was

recorded. On the basis of this time-dependent pressure data,mean values were calculated and used for the calculation ofthe mixture composition.

■ DATA EVALUATIONIn general, the mutual diffusivity is smaller than the thermaldiffusivity, so it can be assumed that the more slowly decayingexponential mode in the recorded CF is related to the mutualdiffusivity, and hence is denoted with τC,c. On the basis of thisassumption, the faster decaying mode can be attributed to thethermal diffusivity and is denoted with τC,t. To access thedecay times attributable to thermal and mutual diffusivity forthe different mixtures, the recorded CFs can be described by atheoretical model according to eq 5. Because of vibrations,particles, incoherent external stray light, or convection in thesample, the CFs are superimposed with additional disturbingsignals (see the example given in Figure 3a). Here, anormalized CF obtained from the experiment for the mixtureof n-C28H58 with CO at a temperature of 498.15 K and apressure of 38.5 bar is shown.When a nonlinear regression is performed to find the

experimental constants and decay times, the theoretical modelhas to take these disturbances into account. In general, thedisturbance term can be expressed by a polynomial up to thethird order. In the present study, a quadratic term wassufficient to describe the disturbing signals well, resulting inthe fit model

This can be shown by performing a multifit procedure(MFP), where all nonlinear regressions are performed on thebasis of a Levenberg−Marquardt algorithm. The MFP servesas a tool to qualify a good description of the CF. On the basisof an initial regression taking into account the complete timerange of the recorded CF, the decay times τC,t

init and τC,cinit are

determined. Both decay times and their uncertainties (k = 2)which result from the nonlinear regression based on eq 6 are

Figure 3. Normalized CF for the mixture with CO as obtained from the experiment (a), residual plot (b), and MFP results (c, d).

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given in Figure 3a. Figure 3b depicts the deviation of thecorrelator data from the initial fit in percent, which does notshow any systematic behavior in this case. After the initialregression, the MFP is applied. In each of the two requiredMFP runs, only one exponential mode is variable while thesecond mode is treated as a constant using the parametersfound in the initial regression. The disturbance term is treatedto be constant as calculated from the initial regression in bothMFP runs. In the example shown in Figure 3, the MFP runfocuses on the more slowly decaying mode, where the fastermode is treated as a constant. During each MFP run, the timerange considered for the regression is varied. For the start-of-fit variation, the data from the first correlator channels areomitted step by step. The last nonlinear regression for thestart-of-fit variation is performed for the time range from0.5τC,c,t

init up to the last correlator datum. This variation resultsin several decay times obtained for the different time rangesconsidered. In Figure 3c, a start-of-fit variation plot isillustrated, showing the deviation of each single decay timefrom the average of all decay times obtained in the start-of-fitvariation in percent. The x-axis represents the considered starttime divided by the individual decay time calculated for thecorresponding time range. This variation procedure is alsoapplied to the end of the data set, which is called end-of-fitvariation. Here, the considered time range starts at thebeginning of the data set while its end varies from 3.5τC,c,t

init to6.5τC,c,t

init . This variation results in an end-of-fit variation plotwhere the x-axis represents the end of the considered timerange divided by the corresponding decay time (Figure 3d).Each single calculated decay time exhibits an uncertaintyresulting from the regression which is represented by theerror bars in the fit-variation plots. In the example in Figure3c,d, the deviations of the single decay times from their meanin the start- and end-of-fit variation do not exceed 1.5%.Because this deviation is within the calculated uncertainty (k= 2) of each single decay time, it can be assumed that theobserved CF is well-described by the theoretical modelincluding the selected disturbance term. Especially the end-of-

fit variation shows systematic deviations in case the chosendisturbance term does not describe the CF sufficiently in thelong-time range. If both MFP runs verify the applicability ofthe selected fit model, the decay times obtained from theinitial regression are considered to be the correct ones and areused for further data processing.The difference between the results obtained by a fit model

using, e.g., a polynomial of second order or a quadraticextension, can be found in the uncertainty of the calculateddecay time. Because of the higher degree of freedom, thepolynomial extension yields an uncertainty larger than that ofthe quadratic extension. The decay times calculatedconsidering the complete time range based on the two fitmodels, however, agree within their combined uncertainties. Acomparison of the variation plots obtained from the MFP forboth fit models also indicates no significant difference. On thebasis of the MFP results and the comparison of several fitmodels using different disturbance terms, a quadraticextension term according to eq 6 was found to describe thedisturbances in all the CFs recorded within the present studybest.The physical origin of the disturbances present in the CFs

was not investigated systematically because it is not of specialinterest in this study. Hence, reduced CFs will be treated inthe following. Here, it is assumed that the disturbance is well-described by the disturbance term, can be subtracted from theoriginal CF, and does not influence the final results. In Figure4a, the reduced CF from the measurement example given inFigure 3a is shown, where the two superimposed exponentialfunctions are indicated separately by the dashed and the chainlines. It is obvious that not only the determined decay timesbut also the variation plots calculated for the reduced CF arein exact accordance with the nonreduced CF (see Figures 3and 4). The uncertainty in the decay times calculated fromthe reduced CF over the complete time range is smaller thanthe uncertainty calculated from the nonreduced CF becausethe degree of freedom in the fit model applied to the reducedCF is smaller.

Figure 4. (a) Reduced normalized CF: ---, exponential mode related to mutual diffusivity; -·-, exponential mode related to thermal diffusivity;residual plot (b); and MFP results (c, d).

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The calculated uncertainties in both decay times τC,c andτC,t strongly depend on the ratios χ = τC,c/τC,t and β = bc /bt.The decay time ratio χ indicates how strongly both decaytimes are separated in time, whereas β describes the amplituderatio between both exponential modes in the recorded CF.A limiting case in conjunction to the amplitude ratio β

could not be found. However, a ratio of less than 0.1drastically increases the uncertainty in the decay timeexhibiting the smaller amplitude. In the present study,amplitude ratios less than 0.1 were found only for themixture with H2O at temperatures of 398.77 and 423.49 K, asshown in the upper panel of Figure 5.

The empirically determined limiting values for theevaluation of two superimposed exponential functions aredecay time ratios larger than about 0.5 but smaller than 2.The nonlinear regression does not recognize two exponentialmodes if χ is within this range; thus, a mode separation is notpossible. Hence, in this range it has to be concluded that thedecay times related to fluctuations in temperature andconcentration match and imply larger uncertainties. Themixtures with CO and H2O did not exhibit decay time ratiossmaller than 2 or larger than 0.5 over the whole temperaturerange (see the lower panel of Figure 5). Estimated decay timeratios of less than 2 but more than 0.5 were found only forthe mixture with H2 at temperatures of 498.15 and 523.15 K,which are indicated in Figure 5 by the open symbols. Here,the procedure described in the following was applied for anestimation of the second present mode.Reduced example CFs recorded for the mixture with H2 at

temperatures from 448.34 to 498.15 K are illustrated in Figure6. The decay times of the two modes approach each otherwith increasing temperature. At a temperature of 498.15 K, amode separation is not possible anymore. However, to access

the decay time connected with the mutual diffusivity, thedecay time which describes the thermal fluctuations wascalculated based on thermal diffusivity data of the pure n-C28H58 measured by DLS and included into the fit model forthe CF of the mixture as a constant. This can be justifiedbecause the thermal diffusivity of the mixtures does notchange significantly compared with the that of pure n-C28H58,see Results and Discussion. The uncertainty in the decay timeof the more slowly decaying mode using the evaluationstrategy given in Figure 6c is underestimated as theuncertainty in the decay time calculated from the thermaldiffusivity of the pure n-C28H58 was not considered.The thermal diffusivity of pure n-C28H58 was measured

before the experiments for the mixtures were carried out. Itcan be assumed that the thermal diffusivity of the mixture isdominated by n-C28H58 because of the very small gasconcentration in the mixture on a mass basis. This assumptionand the measurements for pure n-C28H58 confirm the initialhypothesis that the faster relaxing mode in the recorded CFsof the mixtures denoted by τC,t can be attributed to the decaytime of the hydrodynamic mode governed by thermalfluctuations. The concentration fluctuations characterized bythe mean decay behavior τC,c have therefore to be associatedwith the more slowly decaying mode. The allocation of the

Figure 5. Amplitude (upper panel) and decay time (lower panel)ratios found for the investigated systems: ■, mixture with CO,evaluated; ●, mixture with H2, evaluated; ▲, mixture with H2O,evaluated; ○, mixture with H2, estimated.

Figure 6. Reduced example CFs: ---, exponential mode related tomutual diffusivity; -·-, exponential mode related to thermal diffusivity,recorded for the mixture with H2 at temperatures of 448.34 K (a),473.04 K (b), and 498.15 K (c) including evaluated (a, b) andestimated decay time ratios (c).

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two superimposed exponentially decaying modes to theirphysical origin is additionally confirmed by the temperaturedependence of both decay times. The decay time of the fasterdecaying mode increases with increasing temperature, whilethe decay time of the more slowly decaying mode decreases.This means that in connection with the inverse proportion-ality of the decay times to the corresponding transportproperties, eqs 2 and 3, the fact that the thermal diffusivitydecreases and the mutual diffusivity increases with increasingtemperature helps to identify which mode corresponds towhich kind of fluctuations.Maxwell−Stefan Diffusion Calculations. Long NVT

molecular dynamics (MD) simulations at the experimentaldensity of n-C28H58

25 and on the order of 30 ns wereperformed for the estimation of the Maxwell−Stefan (MS)diffusion coefficient D12

MS of n-C28H58 mixtures with CO, H2,or H2O. Various temperatures were examined where thesimulated systems consisted of 30 n-C28H58 molecules. TheMS diffusivities for H2O correspond to a low H2O molefraction of 0.03 in order to be sure that we are below the

solubility limit. For CO and H2, the mole fraction is equal to0.22. For n-C28H58, a united atom (UA) representation wasused. In particular, the transferable potential for phaseequilibria (TraPPE)26 was employed. TraPPE has shown tobe very accurate for the thermodynamic properties of n-alkanes in pure state and in mixtures over a wide range ofconditions, including the critical point and the range oftemperature and pressure values of importance to the currentproject.Details on the force field functional form and parameters

for the various terms can be found in the literature.19

Standard Lorentz−Berthelot combining rules were used todescribe nonbonded Lennard-Jones interactions between sitesof different types

ε ε ε σσ σ

= =+

and2ij ii jj ij

ii jj

(8)

In these simulations, an increased time step of 2 fs wasused and the equations of motion were integrated using thevelocity Verlet algorithm.27 In all cases, to maintain the

Table 1. Thermal and Mutual Diffusivity Data and Their Uncertainties at the Investigated Temperatures, Mean Pressures,and Estimated Compositions

T (K) p (MPa) xCO,H2,H2O (mol %) a (10−9 m2 s−1) Δa/a D12 (10−9 m2 s−1) ΔD12/D12

n-C28H58

372.83 0.15 − 79.8 0.04 − −397.79 0.16 − 75.6 0.04 − −422.75 0.20 − 71.3 0.04 − −448.21 0.18 − 68.7 0.04 − −472.93 0.23 − 63.4 0.07 − −498.16 0.23 − 59.8 0.08 − −518.15 0.26 − 58.2 0.04 − −

n-C28H58 + CO446.96* 3.52 0.08 65.2 0.10 9.17 0.21448.28 2.03 0.05 65.4 0.07 9.79 0.12471.82* 4.00 0.09 60.8 0.08 11.2 0.12473.28 2.27 0.05 63.0 0.07 12.1 0.11498.15* 3.85 0.08 56.8 0.15 12.8 0.30498.15 3.06 0.06 58.5 0.11 13.9 0.21518.15 3.54 0.06 57.5 0.11 16.4 0.17

n-C28H58 + H2

372.90 3.06 0.04 82.3 0.11 15.0 0.27372.94* 1.77 0.03 80.3 0.05 13.2 0.15397.78 3.20 0.05 75.8 0.05 18.5 0.11397.83* 3.33 0.05 78.7 0.09 19.6 0.14422.72 2.67 0.05 69.7 0.06 21.2 0.28422.72* 3.12 0.05 69.2 0.10 21.1 0.26448.34 3.94 0.07 66.8 0.11 26.9 0.25473.04 3.59 0.07 70.1 0.58 33.4 0.42498.16 3.12 0.07 − − 37.0 0.29523.15 1.58 0.06 − − 47.4 0.19

n-C28H58 + H2O398.77 0.20 − 72.4 0.04 7.40 0.27423.49 0.36 0.15 65.4 0.04 8.43 0.16448.27* 0.29 0.08 66.5 0.01 9.99 0.48449.13 0.63 0.18 66.1 0.03 11.8 0.07473.31* 0.47 0.10 60.8 0.07 14.3 0.14473.15 0.91 0.20 63.6 0.05 14.6 0.16498.15 0.77 0.13 59.4 0.05 18.4 0.29498.15* 0.48 0.08 56.3 0.04 17.1 0.19523.15 0.71 0.09 55.9 0.03 21.6 0.14

*Data not illustrated in Figures 7 and 8.

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temperature fixed at its prescribed value, the Berendsenthermostat28 was used with a coupling constant equal to 0.1ps. Simulations were performed in machines based on IntelXeon CPU 2.8 GHz processor. To decrease the uncertainty inD12

MS calculations, four initially different structures of eachmixture were examined.The Maxwell−Stefan diffusion coefficient is a kinetic factor

and can be determined from the mean-square displacement ofthe center of mass of particles of species 1 (here thesolute)29,30

∑ ∑= + −→∞ = =

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎤⎦⎥⎥D

Nx xmm

x xt

r r t1

6lim

dd

(0) ( )t k

N

kk

N

k12MS

1 2

1

21 2

2

1 1

21 1

(9)

where N is the total number of particles; x1 = N1/N and x2 =N2/N are the mole fractions of species 1 and 2, respectively;N1 and N2 are the number of particles of type 1 and 2,respectively; and m1 and m2 are the masses of the twodifferent types of particles. rk(t) is the position of particle k ofspecies 1 at time t.

■ RESULTS AND DISCUSSIONThe measured thermal and mutual diffusivity data and theiruncertainties for pure n-C28H58 and its mixtures with CO, H2,or H2O over a wide temperature range at the correspondingmean pressures and estimated compositions are listed inTable 1. Each diffusivity value stated in the table represents amean thermal or mutual diffusivity calculated from 12 CFs,which were recorded by two correlators at six differentadjustments of the optical setup. The expanded experimentaluncertainty (k = 2) of each datum is based on the standarddeviation calculated from the 12 individual measurements.The data marked by an asterisk in Table 1 are not illustratedin Figures 7 and 8. The illustrated data are chosen because ofcomparable compositions of the respective mixtures.

The thermal diffusivity of pure n-C28H58 and its mixtureswith CO, H2, or H2O decreases with increasing temperature.This behavior is in agreement with results for pure shorter-chain n-alkanes.31 A significant dependence of the thermaldiffusivity on the composition of the mixtures in the probedcomposition range could not be detected. Our data for puren-C28H58 could be fitted by a third-order polynomial found byapplying a nonlinear regression

= − × + ×

− × + ×

− −

− − −

a T T

T T

( ) [ 5.826 10 1.237 10 ( /K)

3.280 10 ( /K) 2.558 10 ( /K) ] m s

8 9

12 2 15 3 2 1

(10)

In the regression, each thermal diffusivity datum wasweighted inversely to its calculated uncertainty, which isbetween 4 and 8% over the whole temperature range.Compared with that of the pure n-C28H58, the uncertainty inthe thermal diffusivity of the mixtures is larger because thethermal diffusivity had to be extracted from two superimposedexponentially decaying functions in the recorded CFs.Uncertainties in the thermal diffusivity for the mixturesrange from 4 to 15% over the temperature range where amode separation was possible. An uncertainty of more than50% resulted for the mixture with H2 at 473.04 K, as a clearmode separation became difficult because of a decay timeratio of about two. In comparison, uncertainties of less than1% in the thermal diffusivity can be achieved by the DLStechnique for pure fluids or for binary mixtures where thethermal and mutual diffusivity differ by more than one orderof magnitude.32,14 Furthermore, the large uncertainties foundhere result from the challenging sample handling described inthe Experimental Section.Regarding the data evaluation for the mutual diffusivity of

the mixture with H2 at temperatures of 498.15 and 523.15 K,a general trend for the deviation of the thermal diffusivity ofthe pure n-C28H58 from its mixtures is of special interest.Figure 7 shows that the thermal diffusivity data for pure n-C28H58 and the mixtures agree within combined uncertainties.Consequently, it is justified to utilize the data of pure n-C28H58 to evaluate the CFs obtained for the mixturesregarding the mutual diffusivity. The literature data shownin Figure 7 indicated by the open symbols were calculated

Figure 7. Comparison of thermal diffusivity of pure n-C28H58 and itsmixtures: () eq 10; ▼, pure, this study; ▽, pure, calculated basedon literature data;4,20,21 ■, mixture with CO, this study; ●, mixturewith H2, this study; ▲, mixture with H2O, this study.

Figure 8. Comparison of mutual diffusivities: ■, mixture with CO(xCO = 0.05), DLS; ●, mixture with H2 (xH2

= 0.06), DLS; ▲,

mixture with H2O (xH2O = 0.15), DLS; () fit from DLS data; ◨,mixture with CO, MD; ◑, mixture with H2, MD; ◮, mixture withH2O, MD; □, mixture with CO, Rodden et al.;4 ○, mixture with H2,Rodden et al.4

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from the thermal conductivity,20 density,4 and specific heatcapacity at constant pressure21 and exhibit an estimateduncertainty on the order of 10%. This uncertainty representsthe maximum error which was calculated based on thereported uncertainties in density of 0.1% and the specific heatcapacity of 4%. An uncertainty in the thermal conductivitywas not stated in the literature and was estimated to be 3%.Within combined uncertainties, the thermal diffusivitiescalculated from literature data are also in good agreementwith our data, the first thermal diffusivity data directlymeasured for n-C28H58.Our experimental mutual diffusivity data are shown in the

upper panel of Figure 8. The symbols in the figure representthe mean mutual diffusivities obtained by DLS in the presentstudy. The error bars illustrate the uncertainties (k = 2) in themutual diffusivity, which are estimated to range from 4 to30% over the whole temperature range. The mutual diffusivitydata in dependence on temperature for the mixtures with CO,H2, and H2O show an Arrhenius-like behavior. In acorresponding regression, each data point was weightedinversely to its uncertainty. In the lower panel of Figure 8,the deviations of the data obtained by MD simulations andliterature data measured with the Taylor dispersion (TD)technique4 from the Arrhenius-fit of our data are illustrated.The D12

MS values obtained from MD simulations and theiruncertainties are listed in Table 2.

Uncertainties in the experimental mutual diffusivity of lessthan 12% could be achieved when the decay time ratio islarger than two and the amplitude ratio is close to one asobserved for the mixture with CO at temperatures of about473 K. The mixture with H2O at temperatures larger than 450K exhibited amplitude ratios smaller than 0.2. In these cases,the uncertainty in the mutual diffusivity reaches 50% as thethermal fluctuations dominate the CF. For the mixture withH2 at 473.04 K, the limiting condition for the data evaluationstrategy was reached. A decay time ratio equal to two resultedin an uncertainty in the mutual diffusivity greater than 40%.At higher temperatures of 498.15 and 523.15 K, a decay timeratio less than 2 but greater than 0.5 made a mode separation

impossible; thus, the data evaluation strategy based on theknowledge of the thermal diffusivity of the pure n-C28H58 wasused. In this case, the uncertainty in the mutual diffusivity isestimated to be smaller than 30%.The mutual diffusivities investigated by Rodden et al.4 by

TD at infinite dilution differ by less than 15% from the datameasured in this study for the mixture with H2 over the wholetemperature range. For the mixture with CO, a deviationgreater than 25% can be observed only at a temperature of371 K. This can be attributed to the regression applied to ourdata which could be performed only with a startingtemperature of 448 K and obviously cannot sufficientlydescribe the temperature dependence of the mutual diffusivityin the low-temperature region. The mutual diffusivities for themixtures with CO and H2 calculated by MD simulationsdeviate by less than 22% and 20% from the DLS data,respectively. Larger deviations between experimental andsimulated data are observed for the mixture with H2O attemperatures below 473 K. Here, one can conclude that amore realistic force field is needed to account for the H2O−n-alkane interactions, possibly by accounting explicitly for thepolarizability effects. The simulations for the mixtures withCO and H2 were carried out at a mole fraction of 0.22 andfor the mixture with H2O at a mole fraction of 0.03. As thesedata are in good agreement with our data, where thecomposition of all mixtures is estimated to be less than 0.1, apotential concentration dependence of the mutual diffusivityin the investigated composition range cannot be resolved.This result is in agreement with the work of Makrodimitri etal.,19 where mutual diffusivities of the mixtures n-C28H58 withCO and H2 were simulated over a wide concentration range.

■ CONCLUSIONS

It was demonstrated that thermal and mutual diffusivities aresimultaneously accessible by DLS without applying macro-scopic gradients for binary mixtures of n-C28H58 with CO, H2,and H2O. Both transport properties could be determined overa wide range of thermodynamic states with satisfactoryuncertainties. In the temperature range where the super-imposed signals attributable to thermal or mutual diffusivityapproached each other, the uncertainty in both propertiesincreased dramatically. With an improved data evaluationprocedure, however, even in these cases reliable data could beobtained. Although the DLS experiment allows measurementsat different compositions, no dependence of the thermal ormutual diffusivity on the dissolved gas concentration could beevidenced within the measurement uncertainties. This result isin agreement with results from MD simulations of D12

MS for thesame mixtures. For pure n-C28H58, the first directly measuredthermal diffusivity data were obtained at saturation conditionswith an uncertainty of 4 to 8% (k = 2) for temperatures from372 to 518 K.

■ AUTHOR INFORMATION

Corresponding Author*Tel.: +49-9131-85-29789. Fax: +49-9131-85-29901. E-mail:apf@aot.uni-erlangen.de.

NotesThe authors declare no competing financial interest.

Table 2. Mutual Diffusivity Data and Their UncertaintiesObtained from MD Simulations

T (K) D12MS (10−9 m2 s−1) ΔD12

MS/D12MS

xCO = 0.22373 3.78 0.14443 7.96 0.15473 10.5 0.01495 11.0 0.07523 15.4 0.03

xH2= 0.22

373 14.4 0.07443 23.5 0.10473 30.3 0.13495 32.2 0.10523 35.0 0.03

xH2O = 0.03

373 8.03 0.03443 17.1 0.14473 19.4 0.13495 21.1 0.05523 23.0 0.09

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■ ACKNOWLEDGMENTS

This work was financially supported by the German ResearchFoundation (Deutsche Forschungsgemeinschaft, DFG) byfunding the Erlangen Graduate School in Advanced OpticalTechnologies (SAOT) within the German ExcellenceInitiative. Financial support from Shell Global SolutionsInternational BV through a contracted research agreement isgratefully acknowledged.

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